A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions

TL;DR

This paper introduces a new hypothesis related to the distribution of simple zeros of Dirichlet L-functions, showing it follows from the generalized Riemann hypothesis and exploring its implications for zeros and Siegel zeros.

Contribution

The paper proposes a new hypothesis about simple zeros of Dirichlet L-functions and demonstrates its logical consequences assuming the generalized Riemann and Lindel"of hypotheses.

Findings

$RH_{sim}^\

holds for all primitive characters if it holds for one, assuming the generalized Lindel"of hypothesis.

Under the hypothesis, all simple zeros lie on the critical line, ruling out Siegel zeros.

Abstract

For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that $RH_{sim}^\dagger[X]$ (for any $X$) is a consequence of the generalized Riemann hypothesis. Assuming only the generalized Lindel\"of hypothesis, we show that if $RH_{sim}^\dagger[X]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $\chi$ (primitive or not), all simple zeros of $L(s,\chi)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation.